Moving Average/Weighted

Mathematical Definition: Weighted moving average

In technical analysis of financial data, a weighted moving average (WMA) has the specific meaning of weights that decrease in arithmetical progression. In an n-day WMA the latest day has weight n, the second latest n − 1, etc., down to one. These weights create a discrete probability distribution with:

$s(n):=n+(n-1)+\dots +1=\frac{n\cdot (n+1)}{2}$ and $p_t(x)=\{\begin{array}{cc}\frac{n-(t-x)}{s(n)}& \mathrm{f}\mathrm{o}\mathrm{r}0\le t-n\le x\le t,\\ 0& \mathrm{f}\mathrm{o}\mathrm{r}x<t-n\mathrm{o}\mathrm{r}x>t\end{array}$

The weighted moving average can be calculated for $t\ge n$ with the discrete probability mass function $p_t$ at time $t\in {\mathbb{N}}_0:=\{0,1,2\dots ,\}$, where $t=0$ is the initial day, when data collection of the financial data begins and $C(0)$ the price/cost of a product at day $t=0$. $C(x)$ the price/cost of a product at day $x\in {\mathbb{N}}_0$ for an arbitrary day x.

$\text{WMA}(t):=\sum _{x\in T={\mathbb{N}}_0}{p}_t(x)\cdot C(x)=\sum _{x=t-n+1}^t{p}_t(x)\cdot C(x)=\frac{n\cdot C(t)+(n-1)\cdot C(t-1)+\cdots +2\cdot C(t-n+2)+1\cdot C(t-n+1)}{n+(n-1)+\cdots +2+1}$

WMA weights n = 15

The denominator is a triangle number equal to $\frac{n(n+1)}{2}$ which creates a discrete probability distribution by:

$\frac{1}{s(n)}+\frac{2}{s(n)}+\dots +\frac{n}{s(n)}=\frac{1+2+\dots +n}{s(n)}=\frac{s(n)}{s(n)}=1$

The graph at the right shows how the weights decrease, from highest weight at day t for the most recent datum points, down to zero at day t-n.

In the more general case with weights $w_0,\dots ,w_n$ the denominator will always be the sum of the individual weights, i.e.:

$s(n):=\sum _{k=0}^n{w}_k$ and $w_0$ as weight for for the most recent datum points at day t and $w_n$ as weight for the day $t-n$, which is n-th day before the most recent day $t$.

The discrete probability distribution $p_t$ is defined by:

$p_t(x)=\{\begin{array}{cc}\frac{w_{t-x}}{s(n)}& \mathrm{f}\mathrm{o}\mathrm{r}0\le t-n\le x\le t,\\ 0& \mathrm{f}\mathrm{o}\mathrm{r}0\le x<t-n\mathrm{o}\mathrm{r}x>t\end{array}$

The weighted moving average with arbitrary weights is calculated by:

$\text{WMA}(t):=\sum _{x\in T={\mathbb{N}}_0}{p}_t(x)\cdot C(x)=\sum _{x=t-n}^t{p}_t(x)\cdot C(x)=\frac{w_0\cdot C(t)+w_1\cdot C(t-1)+\cdots +w_{n-1}\cdot C(t-n+1)+w_n\cdot C(t-n)}{w_0+\cdots +w_{n-1}+w_n}$

This general approach can be compared to the weights in the exponential moving average in the following section.